Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented.     

Tandem queues with subexponential service times and finite buffers

We focus on tandem queues with subexponential service time distributions. We assume that number of customers in front of the first station is infinite and there is infinite room for finished customers after the last station but the size of the buffer between two consecutive stations is finite. Using (max, +) linear recursions, we investigate the tail asymptotics of transient response times and waiting times under both communication blocking and manufacturing blocking schemes. We also discuss under which conditions these results can be generalized to the tail asymptotics of stationary response times and waiting times.     

A tandem network with a sharing buffer

In this paper, we consider a two-stage tandem network. The customers waiting in these two stages share one finite buffer. By constructing a Markov process, we derive the stationary probability distribution of the system and the sojourn time distribution. Given some constraints on the minimum loss probability and the maximum waiting time, we also derive the optimal buffer size and the shared-buffer size by minimizing the total buffer costs. Numerical results show that, by adopting the buffer-sharing policy, the customer acceptance fraction and the delivery reliability are more sensitive to buffer size comparing with the buffer-allocation policy.     

Scheduling service in tandem queues attended by a single server

Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the i^th queue is an exponentially distributed random variable with rate μ_i. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit     

Tandem queue with group arrivals and no intermediate buffer

We study a system with two single-server stations in series. There is an infinite buffer in front of the first station and no buffer between the two stations. The customers come in groups; the groups contain random numbers of customers and arrive according to a Poisson process. Assuming general service time distributions at the two stations, we derive the Laplace transform and the recursive formula for the moments of the total time spent in the tandem system (waiting time in the system) by an arbitrary customer. From the Laplace transform, we conclude that the optimal order of the servers for minimizing the waiting time in the system does not depend on the group size.     

带负顾客和强占优先权的不耐烦信元排队

考虑带有负顾客的两类信元的强占优先权M/M/1排队系统.两类信元及负顾客的到达过程均为泊松过程.两类信元到达后分别在各自有限的缓冲器内排队,第一类信元较第二类信元有强占优先权,同时第一类信元是不耐烦的.负顾客一对一抵消队尾的第一类信元(若有),若系统中无第一类信元,到达的负顾客就自动消失.负顾客不接受服务.采用矩阵分析的方法得到了两类信元各自的稳态分布,并作了相应的性能分析.     

Expulsion and scheduling control for multiclass queues with heterogeneous servers

We consider an M/M/2 system with nonidentical servers and multiple classes of customers. Each customer class has its own reward rate and holding cost. We may assign priorities so that high priority customers may preempt lower priority customers on the servers. We give two models for which the optimal admission and scheduling policy for maximizing expected discounted profit is determined by a threshold structure on the number of customers of each type in the system. Surprisingly, the optimal thresholds do not depend on the specific numerical values of the reward rates and holding costs, making them relatively easy to determine in practice. Our results also hold when there is a finite buffer and when customers have independent random deadlines for service completion.     

Time-varying tandem queues with blocking: modeling,analysis, and operational insights via fluid models with reflection

In this paper, we develop time-varying fluid models for tandem networks with blocking. Beyond having their own intrinsic value, these mathematical models are also limits of corresponding many-server stochastic systems. We begin by analyzing a two-station tandem network with a general time-varying arrival rate, a finite waiting room before the first station, and no waiting room between the stations. In this model, customers that are referred from the first station to the second when the latter is saturated (blocked) are forced to wait in the first station while occupying a server there. The finite waiting room before the first station causes customer loss and, therefore, requires reflection analysis. We then specialize our model to a single station (many-server fluid limit of the \(G_t/M/N/(N +H)\) queue), generalize it to k stations in tandem, and allow finite internal waiting rooms. Our models yield operational insights into network performance, specifically on the effects of line length, bottleneck location, waiting room size, and the interaction among these effects.     

A Priority Tandem Queue with No Intermediate Buffer

Priority queues are important in modelling and analysis of manufacturing systems, and computer and communication networks. In this paper, a priority tandem queueing system with two stations in series is studied. There is no intermediate buffer between the two stations, and the lack of buffers may cause blocking at the first station. K types of customers arrive at the system according to Poisson processes. The expected delay in the system for each type of customer is obtained when all the customers have the same service time distribution at the second station. Two cases are studied in detail when service times are either all exponentially distributed or all deterministic.     

A tandem GI/PH/1→•/PH/1/0 queue with blocking

Tandem queues are widely used in mathematical modeling of random processes describing the operation of manufacturing systems, supply chains, computer and telecommunication networks. Although there exists a lot of publications on tandem queueing systems, analytical research on tandem queues with non-Markovian input is very limited. In this paper, the results of analytical investigation of two-node tandem queue with arbitrary distribution of inter-arrival times are presented. The first station of the tandem is represented by a single-server queue with infinite waiting room. After service at the first station, a customer proceeds to the second station that is described by a single-server queue without a buffer. Service times of a customer at the first and the second server have PH (Phase-type) distributions. A customer, who completes service at the first server and meets a busy second server, is forced to wait at the first server until the second server becomes available. During the waiting period, the first server becomes blocked, i.e., not available for service of customers. We calculate the joint stationary distribution of the system states at the embedded epochs and at arbitrary time. The Laplace–Stieltjes transform of the sojourn time distribution is derived. Key performance measures are calculated and numerical results presented.     

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.     

Optimal static pricing for a service facility with holding costs

We study a service facility modelled as a single-server queueing system with Poisson arrivals and limited or unlimited buffer size. In systems with unlimited buffer size, the service times have general distributions, whereas in finite buffered systems service times are exponentially distributed. Arriving customers enter if there is room in the facility and if they are willing to pay the posted price. The same price is charged to all customers at all times (static pricing). The service provider is charged a holding cost proportional to the time that the customers spend in the system. We demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. Finally, we consider buffer size as an additional decision variable and show that there is an optimal buffer size level that maximizes profit.     

Optimal admission control for two station tandem queues with loss

We consider a two-station tandem queue with a buffer size of one at the first station and a finite buffer size at the second station. Silva et al. (2013) gave a criterion determining the optimal admission control policy for this model. In this paper, we improve the results of Silva et al. (2013) and also solve the problem conjectured by Silva et al. (2013).     

Analysis of an open tandem queueing network with population constraint and constant service times 1

We consider an open tandem queueing network with population constraint and constant service times. The total number of customers that may be present in the network can not exceed a given value K. Customers arriving at the queueing network when there are more than K customers are forced to wait in an external queue. The arrival process to the queueing network is assumed to be arbitrary. We show that this queueing network can be transformed into a simple network involving only two nodes. Using this simple network, we obtain an upper and lower bound on the mean waiting time. These bounds can be easily calculated. Validations against simulation data establish the tightness of these bounds.     

Optimal pricing for tandem queues with finite buffers

We consider optimal pricing for a two-station tandem queueing system with finite buffers, communication blocking, and price-sensitive customers whose arrivals form a homogeneous Poisson process. The service provider quotes prices to incoming customers using either a static or dynamic pricing scheme. There may also be a holding cost for each customer in the system. The objective is to maximize either the discounted profit over an infinite planning horizon or the long-run average profit of the provider. We show that there exists an optimal dynamic policy that exhibits a monotone structure, in which the quoted price is non-decreasing in the queue length at either station and is non-increasing if a customer moves from station 1 to 2, for both the discounted and long-run average problems under certain conditions on the holding costs. We then focus on the long-run average problem and show that the optimal static policy performs as well as the optimal dynamic policy when the buffer size at station 1 becomes large, there are no holding costs, and the arrival rate is either small or large. We learn from numerical results that for systems with small arrival rates and no holding cost, the optimal static policy produces a gain quite close to the optimal gain even when the buffer at station 1 is small. On the other hand, for systems with arrival rates that are not small, there are cases where the optimal dynamic policy performs much better than the optimal static policy.

     

Scheduling networks of queues: Heavy traffic analysis of a simple open network

We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.     

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.     

Pooling in tandem queueing networks with non-collaborative servers

This paper considers pooling several adjacent stations in a tandem network of single-server stations with finite buffers. When stations are pooled, we assume that the tasks at those stations are pooled but the servers are not. More specifically, each server at the pooled station picks a job from the incoming buffer of the pooled station and conducts all tasks required for that job at the pooled station before that job is placed in the outgoing buffer. For such a system, we provide sufficient conditions on the buffer capacities and service times under which pooling increases the system throughput by means of sample-path comparisons. Our numerical results suggest that pooling in a tandem line generally improves the system throughput—substantially in many cases. Finally, our analytical and numerical results suggest that pooling servers in addition to tasks results in even larger throughput when service rates are additive and the two systems have the same total number of storage spaces.     

Exact Solution to Tandem Fluid Queues with Controlled Input

We consider a tandem fluid system composed of multiple buffers connected in a series. The first buffer receives input from a number of independent homogeneous on-off sources and each buffer provides input to the next buffer. The active (on) periods and silent (off) periods follow general and exponential distribution, respectively. Furthermore, the generally distributed active periods are controlled by an exponential timer. Under this assumption, explicit expressions for the distribution of the buffer content for the first buffer fed by a single source is obtained for the fluid queue driven by discouraged arrivals queue and infinite server queue. The buffer content distribution of the subsequent buffers when the first buffer is fed by multiple sources are found in terms of confluent hypergeometric functions. Numerical results are illustrated to compare the trend of the average buffer content for the models under consideration.